Mathematical Analysis and Optimization for Economists

Mathematics plays a vital role in providing the logical and foundational basis for several other branches of knowledge. This is particularly true in fields such as economics. Attempting to understand economics without a reasonable grip on mathematics is unwise. This has long been felt and worked upon. Consequently, mathematical economics is rich in its literature and a large number of good books on it, both classical (Takayama and Akira 1985; Bushaw and Clower 1957) and modern (Yu 2019; Korn and Luderer 2021; Hoy et al. 2022; Mala 2022a, 2022b) already exist.

The book, Mathematical Analysis and Optimization for Economists, is written with a two-fold intention—on one hand, it envisages making the power and usefulness of contemporary mathematical methodologies evident to the students of economics, and on the other, it attempts at illustrating how these techniques or methodologies could be employed in the solutions of macroeconomic problems.

Chapters 1 through 4 are about mathematical foundations. Mathematical Foundations 1 is about preliminaries and attempts to make sure that the book remains as self-contained as possible. Matrices and determinants are inevitable tools for any analysis course, especially when an analysis book attempts at solving practical problems related to economics. From the very ground-up stuff, such as the definitions of the various types of matrices to more advanced and involved concepts, such as those of linear dependence, linear independence, dimension, rank, convex cones, and quadratic forms, the first chapter of the book, owing to its richness, may (in its own right) be considered a treatise on linear algebra. Mathematical Foundations 2 is about the basics of real numbers. It discusses the important analysis notions of metric spaces, limits of sequences, point-set theory, continuous single-valued functions, and operators. Mathematical Foundations 3 starts with beyond single-valued functions and discusses limits, continuity, and derivatives of single-valued functions. While most contemporary analysis books miss out on the equality of mixed partials, this chapter contains one on Young’s theorem that provides a sufficient set of conditions for the equality of mixed partials f_xy and f_yx. Derivatives of vector-valued functions and quadratic functions are also discussed. Taylor’s formula with integral remainder and Lagrange’s form of the remainder is presented lucidly and with suitable examples. Mathematical Foundations 4 starts with the implicit function theorem that specifies sufficient conditions for an equation or a system of equations to define certain variables as functions of other variables. Chain rule and functional dependence are also explained with a good number of engagingexamples.

After setting the stage in the first four chapters, the book transitions into more advanced topics.

Chapters 5 through 7 discuss the global and local extrema of real-valued functions. Global and Local Extrema of Real-Valued Functions gives a general overview of optimization that is a central common concern to economics, business, and management. More often than not, human activities and endeavors are directed toward some maximization or minimization. This is precisely when mathematics becomes both an indispensable tool and a life-saver. Optimization is an everyday seek. Global Extrema of Real-Valued Functions and Local Extrema of Real-Valued Functions focus, respectively, on what they claim. Special emphasis is laid on bounded functions that are ubiquitous in analysis and the first concern regarding the existence of global extrema. It leads to several beautiful destinations such as the completeness axiom, Weierstrauss’ theorem, and the intermediate value theorem. There is a detailed review of the derivative that comes in extremely handy in identifying and classifying points of maxima, points of minima, and points ofinflection.

Convex and concave real-valued functions are the concern of Chapter 8, Convex and Concave Real-Valued Functions, and Chapter 9, Generalizations of Convexity and Concavity. Convexity and concavity of real functions play a pivotal role in the determination of extreme values. A region is convex if, for any two points in it, the line segment joining the two points wholly lies in it. The chapters discuss, among other things, supergradients of concave and subgradients of convex functions, the differentiability, and extrema of these functions, conjugate functions, and their applications to economics. Quasiconcavity, quasiconvexity (along with their differentiability), pseudoconcavity, and pseudoconvexity have also been discussed thoroughly. Chapter 9 contains two appendices—one on additional thoughts on the chapter’s theorems and another on additional thoughts on differentiable pseudoconcave and pseudoconvexfunctions.

Constrained extrema are discussed in the next three chapters, that is Chapters 10 through 12. Starting with the introduction of Lagrange’s technique, the discussion culminates in an interpretation of the Lagrange multiplier and the generalized technique of Lagrange and a couple of appendices.

Chapter 13 is about Lagrangian saddle points and duality. The main emphasis is on the saddle-point problem. It discusses several necessary and sufficient conditions in this connection.

Chapter 14 discusses generalized concave optimization. It is an advanced topic and may safely be skipped by amateurs and first-timers.

Chapter 15 is about three special kinds of functions—homogeneous, homothetic (functions that can be expressed as monotonic increasing transformations of homogeneous functions), and almost homogeneous. It discusses, besides other concepts, Euler’s theorem for each one of these three types of functions. The chapter throws light on the connection between homogeneity and convexity. There is also a section on homogeneous programming and its economic applications.

Chapter 16 is about envelope theorems. It considers the optimization of a function that depends upon a set of arguments as well as a set of parameters.

Chapter 17 is about Brouwer and Kakutani’s fixed point theorems. It is a miscellaneous collection of fixed-point theorems presented in a variety of forms incorporating varying sets of assumptions. This is one of the most exciting and engaging chapters of the book and contains exciting discussions on simplexes, convex polytopes (sets that can be expressed as convex hulls of finite number of points), and simplicial decomposition and subdivision. The pick of the chapter is a section on the existence of fixed points.

The first 17 chapters of the book deal exclusively with static optimization procedures; however, the subject matter of Chapter 18 is dynamic optimization. Its main focus is on optimal control modeling problems such as the infinite time horizon problem frequently encountered in economic modeling for which the planning period is of infinite duration.

Chapter 19 is a revision of comparative statics which is commonly employed to study changes in supply and demand. It is a methodological concept in economic theory used to study the change from one equilibrium position to another. The chapter begins with a discussion of the fundamental equation of comparative statics and transcends to its economic applications. It discusses the constrained utility maximization problem (that focuses on choosing a commodity bundle that results in maximizing a strictly quasiconcave utility function subject to certain budget constraints), the constrained cost minimization problem (e.g., minimizing advertising expenditures subject to some constraints) usually encountered in managerial decision making and the long-run profit maximizationproblem.

The book contains a good number of references for further reading and a very rich index is appended for a quick look-up. Despite attempts to keep the book self-contained, the book is not an introductory text on mathematical analysis and optimization. It could safely be classified as an intermediate-level textbook. Having said that, it is a very rich source of literature on mathematical analysis and optimization for economists.

Firdous Ahmad Mala
Government Degree College Sopore, Baramulla,Jammu and Kashmir, India